A330411 a(n) is the index of the first 0 term in the rumor sequence with initial 0th term 1 and parameters b = 3 and n.

2, 1, 3, 5, 4, 24, 2, 22, 21, 20, 19, 6, 17, 16, 15, 14, 4, 6, 20, 22, 99, 8, 97, 3, 95, 94, 93, 92, 13, 393, 89, 391, 9, 389, 85, 84, 83, 82, 384, 80, 382, 60, 59, 58, 75, 377, 376, 375, 53, 373, 372, 68, 370, 66, 368, 796, 24, 365, 793, 363, 362, 361, 789, 788, 787, 786, 14, 784

Offset: 1

Author

Petros Hadjicostas, Dec 13 2019

Keywords

• nonn

Comments

"A rumor sequence (running modulus recurrence sequence) is defined as follows: fix integer parameters b > 1 and n > 0. Set z[0] = any integer, and, for k > 0, define z[k] to be the least nonnegative residue of b*z[k-1] modulo (k+n). The rumor sequence conjecture states that all such rumor sequences are eventually 0." [Copied from the comments for A177356]

Examples

			For n = 1, we have z[0] = 1, z[1] = 1, and z[2] = 0, so a(1) = 2.
For n = 2, we have z[0] = 1 and z[1] = 0, so a(2) = 1.
For n = 3, we have z[0] = 1, z[1] = 3, z[2] = 4, and z[3] = 0, so a(3) = 3.
For n = 4, we have z[0] = 1, z[1] = 3, z[2] = 3, z[3] = 2, z[4] = 6, and z[5] = 0, so a(4) = 5.
For n = 5, we have z[0] = 1, z[1] = 3, z[2] = 2, z[3] = 6, and z[4] = 0, so a(5) = 4.
		

Links

• B. Dearden, J. Iiams, and J. Metzger, Rumor Arrays, J. Integer Seq. 16 (2013), #13.9.3.
• B. Dearden and J. Metzger, Running Modulus Recursions, J. Integer Seq. 13(1) (2010), #10.1.6.

Crossrefs

Cf. A177356, A208125.

Programs

Mathematica

For[n=1, n<50, n++, k=0; Clear[z]; z[0]=1; z[k_]:=z[k]=Mod[3z[k-1], k+n];
While[z[k]>0, k++]; Print[k]; ] (* Petros Hadjicostas, Dec 14 2019, modification of a program in A177356 *)

Formula

a(n) = inf{m > 0 | z[0] = 1, z[m] = 0, and z[k] = (3*z[k-1] mod (k + n)) for k = 1..m}.